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## G = C32×Dic3order 108 = 22·33

### Direct product of C32 and Dic3

Aliases: C32×Dic3, C333C4, C324C12, C3⋊(C3×C12), C6.(C3×C6), (C3×C6).7C6, C2.(S3×C32), C6.10(C3×S3), (C3×C6).11S3, (C32×C6).1C2, SmallGroup(108,32)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C32×Dic3
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — C32×Dic3
 Lower central C3 — C32×Dic3
 Upper central C1 — C3×C6

Generators and relations for C32×Dic3
G = < a,b,c,d | a3=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 80 in 52 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C3, C3 [×4], C3 [×4], C4, C6, C6 [×4], C6 [×4], C32, C32 [×4], C32 [×4], Dic3, C12 [×4], C3×C6, C3×C6 [×4], C3×C6 [×4], C33, C3×Dic3 [×4], C3×C12, C32×C6, C32×Dic3
Quotients: C1, C2, C3 [×4], C4, S3, C6 [×4], C32, Dic3, C12 [×4], C3×S3 [×4], C3×C6, C3×Dic3 [×4], C3×C12, S3×C32, C32×Dic3

Smallest permutation representation of C32×Dic3
On 36 points
Generators in S36
(1 15 7)(2 16 8)(3 17 9)(4 18 10)(5 13 11)(6 14 12)(19 33 29)(20 34 30)(21 35 25)(22 36 26)(23 31 27)(24 32 28)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 35 33)(32 36 34)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)
(1 22 4 19)(2 21 5 24)(3 20 6 23)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 32 16 35)(14 31 17 34)(15 36 18 33)

G:=sub<Sym(36)| (1,15,7)(2,16,8)(3,17,9)(4,18,10)(5,13,11)(6,14,12)(19,33,29)(20,34,30)(21,35,25)(22,36,26)(23,31,27)(24,32,28), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33)>;

G:=Group( (1,15,7)(2,16,8)(3,17,9)(4,18,10)(5,13,11)(6,14,12)(19,33,29)(20,34,30)(21,35,25)(22,36,26)(23,31,27)(24,32,28), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33) );

G=PermutationGroup([(1,15,7),(2,16,8),(3,17,9),(4,18,10),(5,13,11),(6,14,12),(19,33,29),(20,34,30),(21,35,25),(22,36,26),(23,31,27),(24,32,28)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22),(25,29,27),(26,30,28),(31,35,33),(32,36,34)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36)], [(1,22,4,19),(2,21,5,24),(3,20,6,23),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,32,16,35),(14,31,17,34),(15,36,18,33)])

C32×Dic3 is a maximal subgroup of   C338(C2×C4)  C338D4  C334Q8  S3×C3×C12  He3⋊C12

54 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 4A 4B 6A ··· 6H 6I ··· 6Q 12A ··· 12P order 1 2 3 ··· 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 ··· 1 2 ··· 2 3 3 1 ··· 1 2 ··· 2 3 ··· 3

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 C3×Dic3 kernel C32×Dic3 C32×C6 C3×Dic3 C33 C3×C6 C32 C3×C6 C32 C6 C3 # reps 1 1 8 2 8 16 1 1 8 8

Matrix representation of C32×Dic3 in GL3(𝔽13) generated by

 3 0 0 0 1 0 0 0 1
,
 3 0 0 0 3 0 0 0 3
,
 1 0 0 0 4 0 0 0 10
,
 1 0 0 0 0 1 0 12 0
G:=sub<GL(3,GF(13))| [3,0,0,0,1,0,0,0,1],[3,0,0,0,3,0,0,0,3],[1,0,0,0,4,0,0,0,10],[1,0,0,0,0,12,0,1,0] >;

C32×Dic3 in GAP, Magma, Sage, TeX

C_3^2\times {\rm Dic}_3
% in TeX

G:=Group("C3^2xDic3");
// GroupNames label

G:=SmallGroup(108,32);
// by ID

G=gap.SmallGroup(108,32);
# by ID

G:=PCGroup([5,-2,-3,-3,-2,-3,90,1804]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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